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It has to do with the fact that the characteristic classes (over the reals) of a principal $G$-bundle have even degree. We can associate Chern-Simons-like theory to each characteristic class of degree $2k$ together with a $G$-bundle $P$ over a manifold of dimension $2k-1$.

To be a bit more technical a Chern-Simons-like form is asssociated to the following data

1. A homogeneous polynomial $\Phi$ of degree $k$ on the Lie algebra of $G$ invariant under the action of $G$ by conjugation.

2. A principal $G$-bundle $P\to M$ over $M$.

3. A pair of connections $\nabla^0, \nabla^1$ on $P\to M$.

The Chern-Weil theory produces two closed forms

$$\Phi(\nabla^0),\Phi(\nabla^1)\in \Omega^{2k}(M)$$

and a form

$$T\Phi(\nabla^1,\nabla^0)\in \Omega^{2k-1}(M),$$

such that

$$d T\Phi(\nabla^1,\nabla^0)= \Phi(\nabla^1)-\Phi(\nabla^0).$$

(For details see Chapter 8 of these notes.)

The transgression form $T\Phi(\nabla^1,\nabla^0)$ is the one used in Chern-Simons theories. It depends on two connections, but usually $\nabla^0$ is some fixed connection.

4 added 14 characters in body

It has to do with the fact that the characteristic classes (over the reals) of a principal $G$-bundle have even degree. We can associate Chern-Simons-like theory to each characteristic class of degree $2k$ together with a $G$-bundle $P$ over a manifold of dimension $2k-1$.

To be a bit more technical a Chern-Simons-like form is asssociated to the following data

1. A homogeneous polynomial $\Phi$ of degree $k$ on the Lie algebra of $G$ invariant under the action of $G$ by conjugation.

2. A principal $G$-bundle $P$.P\to M$over$M$. 3. A pair of connections$\nabla^0, \nabla^1$on$P\to M$. The Chern-Weil theory produces two closed forms $$\Phi(\nabla^0),\Phi(\nabla^1)\in \Omega^{2k}(M)$$ and a form $$T\Phi(\nabla^1,\nabla^0)\in \Omega^{2k-1}(M),$$ such that $$d T\Phi(\nabla^1,\nabla^0)= \Phi(\nabla^1)-\Phi(\nabla^0).$$ (For details see Chapter 8 of these notes.) The transgression form$T\Phi(\nabla^1,\nabla^0)$is the one used in Chern-Simons theories. It depends on two connections, but usually$\nabla^0$is some fixed connection. 3 added 9 characters in body It has to do with the fact that the characteristic classes (over the reals) of a principal$G$-bundle have even degree. We can associate Chern-Simons-like theory to each characteristic class of degree$2k$together with a$G$-bundle$P$over a manifold of dimension$2k-1$. To be a bit more technical a Chern-Simons-like form is asssociated to the following data 1. A homogeneous polynomial$\Phi$of degree$k$on the Lie algebra of$G$invariant under the action of$G$by conjugation. 2. A principal$G$-bundle$P$. 3. A pair of connectyins connections$\nabla^0, \nabla^1$on$P\to M$. The Chern-Weil theory produces two closed forms $$\Phi(\nabla^0),\Phi(\nabla^1)\in \Omega^{2k}(M)$$ and a form $$T\Phi(\nabla^1,\nabla^0)\in \Omega^{2k-1}(M),$$ such that $$d T\Phi(\nabla^1,\nabla^0)= \Phi(\nabla^1)-\Phi(\nabla^0).$$ (For details see Chapter 8 of these notes.) The transgression form$T\Phi(\nabla^1,\nabla^0)$is the one used in Chern-Simons theories. It depends on two connections, but usually$\nabla^0\$ is some canonical fixed connection.

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